Using Isometric Drawings 1 - Nelson

OPEN-ENDEDPathway 1Using Isometric Drawings

Look at the isometric drawing of this cube structure.– Heights are drawn vertically.– Widths and depths are drawn diagonally.– Equal lengths of the structure are equal on the drawing.– The front, which is shaded, is to the left of the closest corner (★).

height

width depth

top

frontfrontrightside

Imagine you are an architect who is going to build a model for your client.

• Build a cube structure using 8 to 10 linking cubes.

• Sketch an isometric drawing of your cube structure on triangle dot paper: – Position your structure so that one of the corners is closest to you

and the front is to the left of that corner.– Draw lines on triangle dot paper to show the layers of cubes.– Label the front and right side on your drawing.

You will need• linking cubes• Triangle Dot Paper

(BLM 16)• a ruler

isometric drawinga sketch on triangle dot paper that shows 3 sides of a 3-D shape from a corner

• All sides of a 3-D shape may not be visible in one isometric drawing because the shape is drawn from only one corner.

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Copyright © 2012 by Nelson Education Ltd.228

Leaps and BoundsUsing Isometric Drawings, Pathway 1

front

right side

e.g.,

• Rotate your cube structure so that a different corner is closest to you. Sketch an isometric drawing from this new perspective.

• Is any part of your cube structure hidden in either of your 2 drawings? If so, explain what is missing.

• Change your cube structure. Sketch the new structure and label it. Make sure there is a part of the structure that is hidden in the isometric drawing.

• A drawing has perspective when one part looks close to you and the rest looks increasingly farther away.

• To avoid confusion about hidden cubes, you can write the total number of cubes you used.

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Leaps and Bounds Using Isometric Drawings, Pathway 1

back

left side

e.g., No, I can see all 9 cubes.

front

right side

e.g., You can't tell if there's a cube in the bottom back corner. I could have used 8 or 9 or 10 cubes.

e.g.,

GUIDEDPathway 1

Jana made this cube structure using 9 linking cubes. She wants to make an isometric drawing of it.

front right side

• You can make an isometric drawing by following these steps:

– Position the cube structure so that the front is to the left of the closest corner (★).

– Draw lines on triangle dot paper to show the layers of cubes. For the bottom layer of cubes that you can see, draw 3 faces for the front and 3 faces for the right side that meet at the closest corner.

– Label the front and the right side on your drawing.

– For the 2nd layer of cubes that you can see, draw 2 faces for the front and 2 faces for the right side that meet at the corner.

– For the tops of the cubes that you can see, draw the top faces.

If you count the cubes you can see in your drawing, there are 8. You can’t see the other cube on the bottom layer.

All the sides of a 3-D shape may not be visible in 1 isometric drawing because it is drawn from only 1 corner of the shape.

Using Isometric Drawings

You will need• linking cubes• Triangle Dot Paper

(BLM 16)• a ruler

isometric drawinga sketch on triangle dot paper that shows 3 sides of a 3-D shape from a corner

front right side

front right side

front right side


Leaps and BoundsUsing Isometric Drawings, Pathway 1

• Jana turned her cube structure around and sketched it again. This time you can see only 6 cubes in her drawing.

To avoid confusion, you can write the total number of linking cubes you used. That way you know how many are hidden.

Try These 1. Assume there are no missing cubes on the bottom layer in these

isometric drawings. Build each cube structure using no more than 10 linking cubes. How many cubes did you use each time?

a)

front right side

________ cubes

b)

front right side

________ cubes

2. Use your cube structure from Question 1b). Add more cubes that can’t be seen in the isometric drawing. How many cubes did you use altogether? Where did you put them?

3. Pentacubes are 3-D shapes made of 5 cubes that are joined.

a) Build 3 of the pentacubes below using linking cubes.

b) Sketch and label an isometric drawing of each model on triangle dot paper.

F LI

P

N

T U

Z

YX

W V

back left side

• You can shade the front faces to help you see the 3-D shape better.

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Using Isometric Drawings, Pathway 1Leaps and Bounds

7 8

e.g., I used 10 cubes by putting 2 more cubes in the back corner.

Q3. e.g., I made pentacubes that look like the letters P, V, T.

4. a) What is the greatest number of linking cubes you could use for building a cube structure that has this isometric drawing?

________ cubes

b) Build the cube structure in part a) using the greatest number of cubes. Then turn the structure around and sketch it from the back.

How many cubes can you see in your drawing?

________ cubes

How many cubes are hidden in the drawing in part a)?

________ cubes

5. Model each cube structure. Then sketch and label an isometric drawing for each.

a) Use 6 linking cubes. The bottom layer is shaped like an L. The only cube in the second layer is placed on the square corner of the L.

b) Use 11 linking cubes.The bottom layer has 3 rows of 2 cubes. The second layer is the same as the first layer, but 1 corner cube is missing.

front right side


Using Isometric Drawings, Pathway 1 Leaps and Bounds

11

8

3

front

right sideright sidefront

e.g., e.g.,

back left side

6. a) Can you always tell how high a structure is by looking at its isometric drawing? Explain your thinking.

b) Can you always see all of the top faces in an isometric drawing? Explain, using an example.

7. Build a cube structure using no more than 10 linking cubes. Sketch and label an isometric drawing of your structure. Write the number of cubes you used.

I used ________ cubes.

8. Is it possible to have different isometric drawings of the same cube structure? Explain your thinking.

We often need to sketch 3-D things in 2-D, and we need ways to make them look 3-D. Isometric drawings are a good way.

FYI


Using Isometric Drawings, Pathway 1Leaps and Bounds

e.g., I think so. The vertical lines in 1 column tell the height.

e.g., No, In the drawing for Question 1b), you can't tell for sure if there are 4 or 5 top faces.

e.g., 10

front

right side

e.g., Yes, because perspectives of the cube structure may be different, like in question 4a and 4b.

OPEN-ENDEDPathway 2

Architects sometimes sketch a 3-D shape from its top, front, and left-side or right-side views to help their clients visualize better. On the top view, they might write the number of cubes that are in each column of the cube structure.

frontfrontright side right

side

top

top

1231 1

Part A

• Build 2 different cube structures, using 10 to 15 linking cubes each time. – Make one structure with a T-shape in the top view.– Make one structure with only 5 cubes visible in the front view.

• How many cubes did you use in each structure?

• For each of your cube structures, sketch and label the front, top, and right side. In the top view, write the number of cubes in each column.

Structure 1: I used ________ cubes.

Structure 2: I used ________ cubes.

Using Different Views

You will need• linking cubes• 2 cm Square Dot

Paper (BLM 14) or 1 cm Square Dot Paper (BLM 15)

• a ruler

• A 3-D cube structure can have some cubes that are hidden.

• A 2-D view shows only the faces of cubes you can see.

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Leaps and BoundsUsing Different Views, Pathway 2

e.g., 12

e.g., 9

2 2 3 3 2

top front right side


3 2 2 1 1

• Look at the cube structures you built. In each one, is the left side the same as the right side? Describe what you see.

• Look at the 3 views you drew for one of your cube structures. Suppose you erased the numbers written on the top view. Can you build a different cube structure that has the same 3 views but contains more cubes or fewer cubes? Try it. Describe what you did.

Part B

This drawing shows the right-side view and top view of a cube structure.

• Build different cube structures that have these views. Describe what you did.

• Do you think that the front views of the structures you just built will be the same? Draw them.

• Not all sides of a 3-D shape are visible in one 2-D view.

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right side top


Leaps and Bounds Using Different Views, Pathway 2

e.g., In my T-structure, the left side is the same as the right side. In my other structure, the left side looks like the right side but it is flipped.

e.g., I can make the T-structure with 11 cubes if I leave out the cube in the middle of the T on the bottom layer. I can't make it with 13 cubes though because the front view changes.

e.g., I built a structure with 7 cubes or 9 cubes with these top views:

e.g., No, here are the front views of my 7-cube structure and my 9-cube structure. They are different.

2 1 1

1 2

2 2 2 2 1

top (7-cube structure) top (9-cube structure)

front (7-cube structure) front (9-cube structure)

GUIDEDPathway 2

You will need• linking cubes• 2 cm Square Dot

Paper (BLM 14) or 1 cm Square Dot Paper (BLM 15)

• a ruler

Adam used 9 cubes to build this cube structure that looks like the shape of an apartment building near his house.

• You can show what Adam’s structure looks like by sketching different views.

– For the top view, look down on the structure to see 5 squares in an L-shape.

– For a side view, look at the left side and see 3 squares high, then 2 squares high.

– For the front view, you can see on the bottom a row of 4 squares, then a 2nd row of 2 squares, and a 3rd row of 1 square.

• To show the number of cubes in the cube structure, you can write a number on the top view that tells the number of cubes in that column.

This is still not enough information to make Adam’s cube structure, because you don’t know the number of cubes to put in each layer.

• You can draw other views.

The right-side view looks like the left-side view, but it’s flipped. The back view looks like the front view, but it’s flipped.

• Try building the structure by using only the right side, left side, and front views. Do you get the same cube structure as Adam’s?

Using Different Views

front

rightside

top view

left-side view

front view

top view with numbers

1123

2

• A 3-D cube structure can have some cubes that are hidden.

• A 2-D view shows only the faces of cubes you can see.

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(oral) e.g., No, I can use 9, 10, 11, 12, or 13 cubes.

Try These1. Circle the drawings below that match views of this cube structure.

Label the views you circled: front, top, left side, right side, or back.

2. Heather built this cube structure using 8 linking cubes.

a) Circle the drawings below that match views of Heather’s cube structure. Label the views you circled: front, top, left side, right side, or back.

b) Can you build a different cube structure that has the same views as the circled views above? Try it. Describe the structure.

3. a) Look at these views for a cube structure. How many linking cubes do you need to build the 3-D shape? (More than 1 answer may be possible.)

________ cubes

b) Explain how you figured out the number of cubes.

front

rightside

frontrightside

front right side top



top no match

no match

no match

front or back, right side or left side

no match

no match

top left side

right side

Yes. e.g., I can make a cube structure with only 6 cubes that has the same left side, right side, and top views.

8, 9, or 10

e.g., I built the shape by choosing 1 view and then rearranging cubes

to match the other views. It took a while to get them right. Then I

counted the cubes.

4. Build a cube structure using the views shown. How many linking cubes did you use?

a) I used ________ cubes.

front top right side

b) I used _______ cubes.

front top left side

5. Chris built this cube structure using 15 linking cubes.

a) The 3 drawings below are supposed to represent views of Chris’s structure, but they are not all correct. Circle the views below that need to be revised.

b) Draw the 3 views correctly.

frontrightside



8

7


3 2 3

1 2 1

1 1 1

6. a) Sketch 3 different views for this cube structure.

b) On the top view, write the number of cubes in each column. Is more than 1 answer possible? Explain.

7. Build 2 cube structures, if possible, for each set of drawings. How many linking cubes did you use each time?

a) I used ________ cubes.

front backtop

b) I used ________ cubes.


c) I used ________ cubes.

top front left side

8. Are 3 views of a structure enough to accurately show a 3-D shape? Explain your thinking.

frontrightside

Graphic designers and architects realize that when you draw a 3-D object in 2-D, it’s important to show it from different perspectives for people to really understand it.

FYI




3 2 1

2

1

e.g., Yes, the 3 could be a 2.

e.g., 12 or 18

e.g., 8

e.g., 28 or 20

e.g., Sometimes 3 views are enough, as in question 7b. It depends on how complicated the structure is. In question 7c, 3 views were not enough.

e.g.,

OPEN-ENDEDPathway 3

You will need• linking cubes• geometric models• blank paper or 2 cm

Square Dot Paper (BLM 14)

• a ruler

A net is a 2-D model of a 3-D shape that shows all of its faces and can be folded to make the 3-D shape.

• Choose 4 different 3-D shapes.– Choose at least one rectangular prism made with linking cubes.– Choose at least one different type of prism.– Choose at least one pyramid.– 2 of your shapes (a pyramid and a prism) should have

the same base.

• Sketch 2 different nets for each shape you chose. Use blank paper or square dot paper.

• Tell what you did. Describe each net.

Shape 1:

Shape 2:

Shape 3:

Shape 4:

Using Nets

• To identify a 3-D shape from a net, look at the number of faces and the shape of each face.

• Faces that touch should be connected on the net. Faces that do not touch should not be connected on the net.

• A 3-D shape is named by the shape of its base. Prisms have 2 bases that are parallel.

pentagon-based prism

base

• Pyramids have a polygon base and triangular faces that meet at a point.

pentagon-based pyramid

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Leaps and BoundsUsing Nets, Pathway 3

e.g., I made a 1-by-8 cube rectangular prism. I placed the 1 square unit faces at either end of a rectangle but they can be placed anywhere on opposite sides of the rectangles.

e.g., I chose a wooden model of a triangular prism. I rolled it on paper and traced around the faces to make the net. It has 2 identical right triangles for ends. For the sides there are 2 rectangles that are the same and 1 bigger rectangle. e.g., I made a pyramid using polydrons. The base is a regular hexagon, connected to 6 triangles that are the same.

e.g., I made a prism using polydrons. The base is a regular hexagon on either end, joined by 6 squares.

(See sample nets in margin.)

Shape 3:

Shape 4:

Shape 2:

Shape 1:

GUIDED

You will need• connecting faces • scissors and tape • linking cubes• 2 cm Square Dot

Paper (BLM 14)• geometric models• a ruler

Pathway 3

Kim made a rectangular prism using 8 linking cubes. You can sketch a net for the prism in different ways.

• You can start by imagining that all the faces of the 3-D shape are folded down.

– Draw two 2-by-4 rectangular faces for the top and bottom, two 1-by-4 rectangular faces for the sides, and two 1-by-2 rectangular faces for the ends.

– Mark the fold lines and cut out the net. Then fold it up to make sure it works.

1-by-2

1-by-4

2-by-4

• You can also sketch a net by rolling the rectangular prism and tracing around the faces.

Using Nets

• A net is a 2-D model of a 3-D shape that shows all of its faces and can be folded to make the 3-D shape.

• Faces that touch should be connected on the net. Faces that do not touch should not be connected on the net.

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Leaps and Bounds Using Nets, Pathway 3

• You can sketch other nets using geometric models. For example, trace the faces of a square-based pyramid, one at a time, and then figure out how to connect them.

• You can use connecting faces to make nets. The model at the right is a pentagon-based prism.

Try These1. Match a net to each 3-D shape. (One net will not have a match.)

a)

b)



2. Sketch a net for this rectangular prism. Mark the fold lines.

3. Does each net below fold into a cube? Explain why or why not.Mark the fold lines on the nets that fold into a cube.

a) d)

b) e)

c) f)


Leaps and Bounds Using Nets, Pathway 3

1 x 7 squares 1 square

No, the 2 ends of the long arms fold over

each other.

No, there is no bottom and 2 tops.

Yes. I drew in the fold lines.

No, it has 4 squares in a block so there will

be overlap.

No, a cube has 6 faces.

Yes, I drew in the fold lines.

4. Write the total number of faces in each 3-D shape. Then, on separate paper, sketch 2 different nets that match the 3-D shape.

a) ________ faces b) ________ faces

pentagon-based pyramid hexagon-based prism

c) Explain how you made your nets for either part a) or part b).

5. On separate paper, sketch nets for 2 different 3-D shapes using any number of the polygons below. Label the base on each net. Write the total number of faces in each 3-D shape.

Shape 1: ________ faces Shape 2: ________ faces

A

BC D E F

6. How do you know when a net will fold into a particular 3-D shape?

7. a) A net is made up of 8 faces. What might the 3-D shape be?

b) A net includes triangles and rectangles. What might the 3-D shape be?

Nets are useful when you want to determine the surface area of a 3-D shape.

FYI



6 8

e.g., I used polydrons: a regular pentagon and 5 identical triangles (1 for each side of the pentagon). I connected all the edges to make

sure the pieces fit together. Then I took the shape apart again in 2 different ways, and traced around the faces to make the nets.

e.g., 5 e.g., 5

e.g., Count the number and type of faces on the 3-D shape to see if they match the faces on the net. Match the shapes that connect on the 3-D shape to the faces that connect on the net.

a 7-sided pyramid or a hexagon-based prism

a triangular prism or a rectangle-based pyramid

Q5. e.g., Shape 1

Q5. e.g., Shape 2

pentagon base

regular hexagon base

Q4a) e.g.,

Q4b) e.g.,

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Using Isometric Drawings 1 - Nelson